Full Navier-Stokes equations were conducted to determine the feasibility of automating the control of wave instabilities within a flat plate boundary layer with sensors, actuators, and a spectral controller. The results indicate that a measure of wave cancellation can be obtained for small and large amplitude instabilities without feedback; however, feedback is required to optimize the control amplitude and phase for exact wave cancellation

Erlebacher, G.

Gunzburger, M. D.

Hussaini, M. Y.

Joslin, R. D.

Nicolaides, R. A.

English

NASA Technical Reports Server

NASA-C_-?03336 S/_,
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- _ ___, !_.,_
Active Controlof Boundary-LayerInstabilities:
Use _fSensorsand Spectral Controller
R. D. Joslin, R. A. Nicolaides, G_Erlebacher,
M. Y. Hussaini, M. D. Gunzburger
Reprinted from
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Volume33,Number8, Pages1521-1523
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AIAA JOURNAL,VOL.33, NO. 8: TECHNICAL NOTES 1521
Active Control of
Boundary-Layer Instabilities:
Use of Sensors and
Spectral Controller
R. D. Joslin,* R. A. Nicolaides, t G. Erlebacher,*
M. Y. Hussaini, _ and M. D. Gunzburger 1
NASA Langley Research Center,
Hampton, Virginia 23681-0001
Introduction
HIS second study focuses on the suppression of instabilitygrowth using an automated active-control technique. This au-
tomated approach is the next logical step based on previous exper-
imental and computational studies reviewed by Joslin et al. t and
Thomas, 2 in which the control was in the form of wave cancella-
tion. The wave-cancellation method assumes that a wavelike dis-
turbance can be linearly canceled by introducing another wave that
has a similar amplitude but that differs in phase. Both experimental
and computational results have demonstrated that two-dimensional
Tollmien-Schlichting (TS) waves can be superposed upon two-
dimensional waves in such a way as to reduce the amplitudes in the
original waves under the presumption of wave cancellation. Joslin
et al. t have definitively shown that flow control by wave cancellation
is the mechanism for the observed phenomena. Three simulations
were performed in their computational study to demonstrate the
wave-cancellation concept. The first simulation obtained the evolu-
tion of a two-dimensional instability generated by periodic suction
and blowing forcing, the second simulation yielded an instability
caused by a suction and blowing actuator in the absence of and
downstream of the forcing used in the first simulation, and the third
simulation computed the evolution of a disturbance resulting from
both forcing and actuator suction and blowing (wave-cancellation
test case). Joslin et al) showed that the superposition of the first and
second simulation results exactly matched the wave-cancellation
simulation results.
Based on the wave-cancellation assumption, the evolution and au-
tomated control of spatially growing two-dimensional disturbances
in a flat-plate boundary layer are computed. Although the present
active-control approach is demonstrated here for a two-dimensional
instability test case, the ultimate goal of this line of research is to
introduce automated control to external flow over an actual aircraft
or to any flow that has instabilities that require suppression.
The nonlinear computations consist of the integration of the sen-
sots, actuators, and controller as follows: the sensors will record
the unsteady pressure or shear on the wall; the spectral analyzer
(controller) will analyze the sensor data and prescribe a rational
output signal; the actuator will use this output signal to control the
Received Nov. 26, 1994; revision received Jan. 31, 1995; accepted for
publication Jan. 31, 1995. Copyright © 1995 by the American Institute of
Aeronautics and Astronautics, Inc. No copyright is asserted in the United
States under Title 17, U.S. Code. The U.S. Government has a royalty-free
license to exercise all rights under the copyright claimed herein for Govern-
mental purposes. All other rights are reserved by the copyright owner.
*Research Engineer, Fluid Mechanics and Acoustics Division. Member
AIAA.
tConsultant, Institute for Computer Applications in Science and Engi-
neering; currently Professor, Department of Mathematics, Carnegie Mellon
University, Pittsburgh, PA 15213.
tResearch Fellow, Institute for Computer Applications in Science and
Engineering. Member AIAA.
_tDirector, Institute for Computer Applications in Science and Engineer-
ing. Member AIAA.
I Consultant, Institute for Computer Applications in Science and Engineer-
ing; currently Professor, Department of Mathematics, Virginia Polytechnic
Institute and State University, Blacksburg, VA 24061.
Laminar Instability Control Laminar
Fig. 1 Schematic of active control with wave cancellation.
disturbance growth and stabilize the instabilities within the laminar
boundary layer. This scenario is shown in Fig. 1. Although a closed-
loop feedback system could be implemented (using an additional
sensor downstream of the actuator) to fully automate the control
and to lead to an exact cancellation of the instability, the feedback
will not be introduced here due to the added computational expense
of the iterative procedure.
Numerical Techniques
The nonlinear, unsteady Navier-Stokes equations are solved by
direct numerical simulation (DNS) of disturbances, which evolve
spatially within the boundary layer. The spatial DNS 3'4 approach
involves spectral and high-order finite difference methods and
a three-stage Runge-Kutta method s for time advancement. The
influence-matrix technique is employed to solve the resulting pres-
sure equation. 6'7 Disturbances are forced into the boundary layer
by unsteady suction and blowing through a slot in the wall. At
the outflow boundary, the buffer-domain technique of Streett and
Macaraeg s is used.
The equations are nondimensionalized with the freestream ve-
locity Uoo, the kinematic viscosity u, and the inflow displacement
thickness 3c]. The Reynolds number becomes R = U_8_/v, and the
frequency is _ = w*8_/Uoo.
Control Method
Here, the term "controller" refers to the logic that is used to
translate sensor-supplied data into a response for the actuator, based
on some control law. For the present study, a spectral controller
requires a knowledge of the distribution of energy over frequencies
and spatial wave numbers. For this automated controller system, a
minimum of two sensors must be used to record either the unsteady
pressure or unsteady shear at the wall. By using Fourier theory, this
unsteady data can be transformed via
f(w) = f(t)e -i'°' dt (1)
oo
where f(t) is the signal and co is the frequency. This transform
yields an energy spectrum that indicates which frequencies exist in
the signal and how much relative energy each frequency contains.
The largest Fourier coefficient indicates the frequency that will
be used to control the instability, although the largest growth rate
can be used instead of largest coefficient. The information from the
two sensors is used to obtain estimates of both spatial growth rates
and phase via the relation
1 dA
a = ---- (2)
Adx
This temporal and spatial information is then substituted into the
assumed control law, or wall-normal velocity boundary condition,
v, (x, t) = v,, × p'u,exp[i(w + dpr)t + ax,] + c.c. (3)
where p_, is the complex pressure (or shear) for the dominant fre-
quency mode (or largest growth-rate mode) at the first sensor, w
is the dominant mode determined from Eq. (1), _bs is the phase
1522 AIAA JOURNAL, VOL. 33, NO. 8: TECHNICAL NOTES
information, t is the time, ot is the growth-rate and wave number
information calculated from Eq. (2), and x, is the distance between
the first sensor and the actuator. Because the sensor information can
be used only to approximate the actuator amplitude and temporal
phase, vw and ¢,_ are parameters that must be optimized to obtain
exact wave cancellation. This may be accomplished through, for
example, a gradient descent algorithm. (Here, no attempt was made
to demonstrate exact wave cancellation.)
This control law is used only for this feasibility study. Aspects
of formal optimal-control theory and artificial neural-network algo-
rithms are currently being tested by the authors for use in a subse-
quent study.
Numerical Experiments
For this study, we are not concerned with the method by which
disturbances are ingested into the boundary layer; the underlying
assumption here is that natural transition involves some dominant
disturbances that can be characterized by waves. In a subsequent
study, we will explore controlling transition consisting of either
random unsteady or three-dimensional nonlinear and arbitrary in-
stabilities. Here, the instabilities are assumed to be characterizable
by discrete frequencies within the spectrum.
For the computations, the grid has 661 streamwise and 61 wall-
normal points. The far-field boundary is located 758_ from the wall,
and the streamwise distance is 3088_ from the inflow, which is equal
to approximately 11 TS wavelengths. The disturbance frequency is
Fr = _o/R x 106 = 86, and the Reynolds number is R = 900
at the inflow. (The streamwise range of the computations and the
relative sensor and actuator are located within the unstable region
of the linear stability neutral curve.) A time-step size of 320 steps
per period is chosen for the three-stage Runge-Kutta method. To
complete a two-dimensional, simulation, 0.9 h on the Cray Y-MP
are required with a single processor. (Refer to Ref. 3 for details on
accuracy issues with grid refinement.)
For this study, the disturbance forcing slot has a length 5.13&_
and is centered 23.108_ downstream of the computational inflow
boundary. The first sensor is located 57.883_ downstream of the
inflow, and the second sensor is located 2.338_ downstream of the
first sensor. The actuator has a slot length 4.673_ and is located
77.94_ downstream of the inflow boundary. These separation dis-
tances were chosen arbitrarily for this demonstration. Ideally, the
forcing, sensors, and actuator should have a minimal separation dis-
tance to improve the accuracy of the sensor information provided to
the actuator.
A small-amplitude disturbance (vf = 0.01%) is forced at the in-
flow and controlled via the automated control law without feedback.
Figure 2 shows the TS wave amplitudes with downstream distance
for the present results compared with the control case (v,, = 0.9v f;
_bt = 1.27r/to) of Joslin et al. t and the uncontrolled wave. The
xl0 -a
2.0 i
-e FFT
n 17 _0.gv!
o v_=0.O
1.5 /
ul 1.o
0.5 _
0.0 , I I
0.9 1.0 1,1 .2 x1-0 3
R
Fig. 2 Active control of small-amplitude TS waves in flat-plate bound-
ary layer.
0.05 I
_,_ FFT
n vu,= 0.9'u.f
e v_=0.0
q r
/
0.04 /
ut 0.03
J
0.02
0.01
O. O0 , J , i ,
0.9 1.0 1.1 1.2 xlO 3
R
Fig. 3 Active control of large.amplitude TS waves in fiat-plate bound-
ary layer.
present results demonstrate that a measure of wave cancellation can
be obtained from the automated system before initiating feedback;
however, feedback is necessary to optimize the control amplitude
and phase for exact cancellation of the disturbance.
Next, the evolution and control of a large-amplitude disturbance
(v/ = 3%) is studied. This large-amplitude case excites harmon-
ics and a mean-flow distortion component with much smaller am-
plitudes than the fundamental-mode amplitude. Figure 3 shows
comparisons for the control of the large-amplitude case. Again, the
automated control can clearly obtain a degree of wave cancellation
for large-amplitude instabilities without optimization and without
causing the harmonics to prematurely grow. These test cases demon-
strate that automated control can be effective with the presumption
of discrete frequency instability waves.
Conclusions
Full Navier-Stokes simulations were conducted to determine the
feasibility of automating the control of wave instabilities within
a flat-plate boundary layer with sensors, actuators, and a spectral
controller.
The results indicate that a measure of wave cancellation can be ob-
tained for small- and large-amplitude instabilities without feedback;
however, feedback is required to optimize the control amplitude and
phase for exact wave cancellation.
This study is only the second in a series aimed at suppressing
the instabilities that lead to transition within an otherwise laminar
boundary layer with unsteady flow control. Follow-on research will
focus on coupling optimal control theory with the Navier-Stokes
equations to devise a control methodology without distinct control
laws. This methodology focuses on the minimization of the wall
shear at a prescribed region downstream of the actuator. This flow
control could lead to suppression of arbitrary instabilities in a lam-
inar boundary layer, drag reductions in a turbulent boundary layer,
or enhanced lift by separation control.
Acknowledgment
This research was supported by NASA under Contract NAS 1-
10801 while the authors (except the first author) were in residence at
the Institute for Computer Applications in Science and Engineering,
NASA Langley Research Center, Hampton, Virginia.
References
lJoslin, R. D., Erlebacher, G., and Hussaini, M. Y., "Active Control of
Instabilities in Laminar Boundary-Layer Flow. An Overview," Journal of
Fluids Engineering (submitted for publication); see also Inst. of Computer
Applications in Science and Engineering, ICASE Rept. 94-97, Hampton,
VA, Dec. 1994.
2Thomas, A. S. W., "Active Wave Control of Boundary-Layer Transition,"
_scous Drag Reduction in Boundary Layers, edited by D. M. Bushnell and
J. N. Hefner, Vol. 123, Progress in Astronautics and Aeronautics, AIAA,
Washington, DC, 1987. pp. 179-202.
AIAA JOURNAL, VOL. 33, NO. 8: TECHNICAL NOTES 1523
3joslin, R. D., Streett, C. L., and Chang, C.-L., "Validation of Three-
Dimensional Incompressible Spatial Direct Numerical Simulation Code--A
Comparison with Linear Stability and Parabolic Stability Equations Theories
for Boundary-Layer Transition on a Flat Plate," NASA TP-3205, July 1992.
4joslin, R. D., Streett, C. L., and Chang, C.-L., "Spatial DNS of Boundary-
Layer Transition Mechanisms: Validation of PSE Theory" Theoretical Com-
putational Fluid Dynamics, Vol. 4, No. 6, 1993, pp. 271-288.
5Williamson, J. H., "Low-Storage Runge-Kutta Schemes," Journal of
Computational Physics, Vol. 35, No. I, 1980, p. 48.
6Danabasoglu, G., Biringen, S., and Streett, C. L., "Spatial Simulation
of Instability Control by Periodic Suction and Blowing," Physics of Fluids,
Vol. 3, No. 9, 1991, pp. 2138-2147.
7Streett, C. L., and Hussalni, M. Y., "A Numerical Simulation of the
Appearance of Chaos in Finite-Length Taylor-Couette Flow," Applied Nu-
merical Mathematics, Vol. 7, No. 1, 1991, p. 41.
SStreett, C. L., and Macaraeg, M. G., "Spectral Multi-Domain for Large-
Scale Fluid Dynamic Simulations," Applied Numerical Mathematics, Vol.
6, No. 1-2, 1989/1990, p. 123.


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